Difference between revisions of "Saturated models"

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In a recent paper Nurkhaidarov and  Schmerl 2011 proved the following: Let κ be regular, uncountable, and such that κ<κ=κ. For each completion TPA let  MκT be the saturated model of T of cardinality κ. There is a set T of completions of PA, such that |T|=20 and for all T,TT, if TT, then Aut(MκT)Aut(MκT). The following question is left open: Are there T and T such that TT and Aut(MκT)Aut(MκT)?
 
In a recent paper Nurkhaidarov and  Schmerl 2011 proved the following: Let κ be regular, uncountable, and such that κ<κ=κ. For each completion TPA let  MκT be the saturated model of T of cardinality κ. There is a set T of completions of PA, such that |T|=20 and for all T,TT, if TT, then Aut(MκT)Aut(MκT). The following question is left open: Are there T and T such that TT and Aut(MκT)Aut(MκT)?
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== Elementary cuts ==
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A cut in a saturated model is balanced if its upward and downward cofinalities are equal to the cardinality of the model. Since there are continuum theories of pairs (N,M), where NPA and MPA, and N is an elementary end extension of M, it follows that  for every saturated model N there are continuum non elementarily equivalent, hence nonisomorphic, pairs (N,M), where M is an elementary balanced cut of N. For a saturated N, what is the maximum number of such pairs?
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For an almost complete classification of elementary cuts in saturated models of PA see James Schmerl  ''Elementary cuts in saturated models of Peano arithmetic''. Notre Dame J. Form. Log. 53 (2012), no. 1, 1–13,

Revision as of 09:24, 18 January 2013

The automorphism group

In a recent paper Nurkhaidarov and Schmerl 2011 proved the following: Let κ be regular, uncountable, and such that κ<κ=κ. For each completion TPA let MκT be the saturated model of T of cardinality κ. There is a set T of completions of PA, such that |T|=20 and for all T,TT, if TT, then Aut(MκT)Aut(MκT). The following question is left open: Are there T and T such that TT and Aut(MκT)Aut(MκT)?


Elementary cuts

A cut in a saturated model is balanced if its upward and downward cofinalities are equal to the cardinality of the model. Since there are continuum theories of pairs (N,M), where NPA and MPA, and N is an elementary end extension of M, it follows that for every saturated model N there are continuum non elementarily equivalent, hence nonisomorphic, pairs (N,M), where M is an elementary balanced cut of N. For a saturated N, what is the maximum number of such pairs?

For an almost complete classification of elementary cuts in saturated models of PA see James Schmerl Elementary cuts in saturated models of Peano arithmetic. Notre Dame J. Form. Log. 53 (2012), no. 1, 1–13,